Imaginary number challenge (3)

Algebra Level 4

( ( ( ( i 2 i ) 3 i ) 4 i ) ⋯ ) 2016 i \large\left(\left(\left(\left(i^{2i}\right)^{3i}\right)^{4i}\right)^{\cdots}\right)^{2016i}

If the principal value of the expression above can be written in the form exp ⁡ ( x ! π y ) \exp\left(\dfrac{x!\pi}y\right) where x x and y y are positive integers, find x + y x+y .

Details and Assumptions:

  • Every complex number z z can be written in the form z = r e i θ + 2 k π z=re^{i\theta+2k\pi} for some real r , θ r,\theta and all k k such that − π < θ ≤ π -\pi\lt \theta\leq\pi . The value when k = 0 k=0 is called the principal value of z z .
  • i = − 1 i=\sqrt{-1} denotes the imaginary unit.
  • exp ⁡ ( x ) = e x \exp(x)=e^x where e e denotes the Euler's number .
  • n ! n! denotes the factorial function. For example, 8 ! = 1 × 2 × 3 × ⋯ × 8 8! = 1\times2\times3\times\cdots\times8 .


The answer is 2018.

Tommy Li by Tommy Li , 22, Hong Kong

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1 solution

Tommy Li
Jun 23, 2016

e π 2 i = cos ⁡ ( π 2 ) + i sin ⁡ ( π 2 ) e^{\frac{\pi}{2}i}=\cos(\frac{\pi}{2})+i\sin(\frac{\pi}{2})

i = e π 2 i i=e^{\frac{\pi}{2}i}

i 2 i = ( e π 2 i ) 2 i = e 1 2 ( 2 ! ) ( π ) ( i 2 ) = e − π i^{2i}=(e^{\frac{\pi}{2}i})^{2i}=e^{\frac{1}{2}(2!)(\pi)(i^{2})}=e^{-\pi}

( i 2 i ) 3 i = ( e − π ) 3 i = e 1 2 ( 3 ! ) ( π ) ( i 3 ) = e − 3 π i (i^{{2i})^{3i}}=(e^{-\pi})^{3i}=e^{\frac{1}{2}(3!)(\pi)(i^{3})}=e^{-3\pi i}

( ( i 2 i ) 3 i ) 4 i = ( e − 3 π i ) 4 i = e 1 2 ( 4 ! ) ( π ) ( i 4 ) = e 12 π ((i^{{2i})^{3i}})^{4i}=(e^{-3\pi i})^{4i}=e^{\frac{1}{2}(4!)(\pi)(i^{4})}=e^{12\pi}

Observe the pattern :

( ( ( ( i 2 i ) 3 i ) 4 i ) … ) n i = e 1 2 ( n ! ) ( π ) ( i n ) \large \left(\left(\left(\left(i^{2i}\right)^{3i}\right)^{4i}\right)^{\ldots}\right)^{ni}=e^{\frac{1}{2}(n!)(\pi)(i^n)}

( ( ( ( i 2 i ) 3 i ) 4 i ) … ) 2016 i = e 1 2 ( 2016 ! ) ( π ) ( i 2016 ) \large \left(\left(\left(\left(i^{2i}\right)^{3i}\right)^{4i}\right)^{\ldots}\right)^{2016i}=e^{\frac{1}{2}(2016!)(\pi)(i^{2016})}

= e 2016 ! π 2 \large = e^{\frac{2016!\pi}{2}}

⇒ x + y = 2016 + 2 = 2018 \Rightarrow x+y=2016+2=2018

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